Answer to Question #146037 in Optics for Water

Question

A grating (with 15,000 lines per inch) is illuminated by sodium light. The grating spectrum is observed on the focal plane of a convex lens of focal length 10 cm. Calculate the separation between the D1 and D2 lines of sodium (The wavelengths of D1 and D2 lines are 589.0 and 589.6 nm respectively).

Answer 1

The condition of diffraction is given by the following equation.

$dsinθ = mλ$

d – is the split separation

m – is the order of diffraction

λ is the wavelength

θ is the diffraction angle

The slit separation d is the reciprocal of total number of lines N on the grating.

$d = \frac{1}{N}$

$d = \frac{1}{15000 \;lines/inch}$

$= 6.666 \times 10^{-5} \;inch(\frac{2.54\;cm}{1 \;inch})$

$= 1.69 \times 10^{-4} \;cm$

$θ = sin^{-1}(\frac{mλ}{d})$

m =1

λ = 598 nm

$θ = sin^{-1}(\frac{1 \times 598.0 \times 10^{-7}}{1.69 \times 10^{-4}}) = 20.7 º$

$∆θ = \frac{m}{dcosθ}∆λ$

∆λ = 589.6 – 589.0 = 0.6 nm

$∆θ = \frac{1}{1.69 \times 10^{-4}cos(20.7)}0.6 \times 10^{-7} = 3.795 \times 10^{-4} \;radians$

The separation between the lines of sodium is equal to the product of focal length and angular width.

∆d = f∆θ

f = 10 cm

$∆d = 10 \times 3.795 \times 10^{-4} = 0.03795 \; mm$

The separation between the lines of sodium is 0.038 mm.